Performance evaluation of Call-center with call redirection. Ocena performanc klicnega centra s preusmerjanjem klicev

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1 Elektotehiški vestik 74(-): Electotechical Review: Lublaa Sloveia Pefomace evaluatio of Call-cete with call ediectio Vladimi Efimushki Dago Žepi Cetal Sciece Reseach elecommuicatios Istitute(ZNIIS) 8 st poezd Peova polya 4 Moscow Russia Iskatel Lublaska c.4a 4 Ka Sloveia ef@ziis.u zepic@iskatel.si Aotatio. he obect of ivestigatio is a aalytical model of a Call-cete fuctioig with a taffic distibutio (call ediectio) mechaism. Call-cete fuctioig is descibed by the Makov pocess. A solutio fo statioay distibutio is foud ad expessios fo the mai pefomace chaacteistics fo the Call-cete fuctioig ae give. Keywods: Call-cete call ediectio aalytical model pefomace evaluatio Ocea pefomac klicega ceta s peusmeaem klicev Povzetek. Obekt aziskave e aaliti'i model klicega ceta z vgaeim mehaizmom za peusmeae klicev. Delovae klicega ceta sva opisala z Makovskim pocesom. Za model sva podala ešitev za stacioao poazdelitev pikazala sva tudi ešitve za glave pefoma'e kaakteistike. Klue besede: klici cete peusmeae klicev aaliti'i model ocea pefomac available agets i the G() goup. If all agets i goups ae occupied the call is lost ad it wo t be tasfeed agai. g i g i C µ Model Desciptio Developmet of Call-cete fuctioig schemes has give ise to ivestigatios i ew call-maagemet schemes chages i the umbe of agets ad call fowadig [-4]. Let s examie a Call-cete that cosists of two goups of agets: G with capacity 5 ad G with capacity 5 (see Fig.). he icomig calls to G() ae peseted by Poisso aivals with the ( ) desity. he times of call pocessig by ay aget of G() ae idepedet adom vaiables distibuted accodig to the expoetial law with the µ ( µ ) paamete. A icomig call to G() call is ediected to G() fo pocessig with a pobability of g i ( g ) which depeds o i ( umbe of busy agets i G(). Othewise the call is pocessed by its ow goup G() with additioal pobability g i ( g ) if thee ae Received 8 Novembe 6 Accepted 6 Decembe 6 g g Figue. A geealized aalytical model of a Call-cete with call ediectio. ( t be a adom vaiable descibig the umbe of calls beig pocessed i G ad G espectively. We examie the Makov pocess { ( t) ( t) t } with the state space X = X X X = {... C } X = {... C}. As all states of the pocess commuicate ad thei umbe is fiite statioay pobability distibutio = lim p ( t) t p i ( t) = P{ ( t) = ( t) = t } exists [5] ad it ca be obtaied via a system of equilibium equatios of X dimesio ad X ak of the fom Let { t) ( )} C µ

2 8 Efimushki Žepi< + p A = () ad omalizig coditio p = () whee p = ( p p... p ) p ( ) )... C )) C i =. Matix A is of the followig fom: A D B A D B A D A = K K K K K K B C AC DC BC A C whee A = a i = C C B = b i = C C D = d i = C C ad is a zeo quadatic matix of the C ode. Let = g + g = g + g. Foms of elemets of the A B D matices ae: u( 5 ) u( 5 µ µ = i = C a i = = i + i = C µ = i i = C othewise = i = C b µ = othewise = = = i C d i othewise. x > Hee u( x) = x. he solutio of the system i Eq. () is peseted i the fom of p p ~ + = A whee A = µ ~ ~ A = ( D + AA) = µ ~ ~ ( A D + AA ) = C. ( + ) µ he vecto p is detemied though the equatio system ~ ~ p ( AC D + A A ) = C C C ad the omalizig coditio i Eq. () obtaiig the fom C ~ p ( I + A ) =. = It is also possible to fid the solutio with othe methods fo example with LU-decompositio. he model povides a oppotuity to examie ad ivestigate diffeet schemes of taffic ediectio. his is doe ust by settig the coespodig g i ad g pobabilities distibutio. If g i = ad g = we get a stadad model of the Call-cete fuctioig without taffic ediectio (model m ). If g i = i = C g C = ad g = = C g C = we get a model with patial taffic ediectio fo the cases of G ad G oveload (model m ). he above poposed geealized model (model 3 m 3 ) eables ivestigatio of diffeet combiatios of ediectio mechaisms. We will use otatio m l _ m k which meas that model ml is implemeted i G ad model mk is implemeted i G l k = 3. Model m l _ ml coespods to the case of a homogeeous model ad model l k coespods to the case of a heteogeeous model. Pefomace Evaluatio Chaacteistics Now that we kow the pobability distibutio of a umbe of busy agets i G ad G goups we ca calculate the ecessay pefomace chaacteistics chaacteizig the effectiveess of the models ude ivestigatio. Let be the loss pobability i the system. I 3 3_ ad 3_3 models call loss occus due to occupatio of all C ad C agets i G ad G i.e. = p ( C C ). I the _ model the loss pobability is calculated accodig to the followig fomula

3 Pefomace evaluatio of Call-cete with call ediectio 8 = ( + ) + C C whee = C ( = C) ) is the statioay pobability of the call loss i G (G). I _ ad _3 models the call loss occus i case of occupatio of all G agets. hus the loss pobability fo the _ ad _3 models is calculated accodig to the fomula C = p ( C ( + g i ) +. Similaly the loss pobability fo the _ ad 3_ models is C = ( ) ( ) + p i C g i +. Let p be the pobability of a idle Call-cete i.e. the pobability of absece of ay call (it is valid fo all models). hus p is equal to the pobability of ( t) = { ( t) ( t ) t } pocess beig i ( ) state. he aveage umbe of busy agets Q ( Q ) i G (G) goup espectively is detemied i the followig way: C C C C Q = i ( Q = ). 3 Case studies Let s coduct a umeical aalysis of the loss pobability fo the ivestigated model ad its alteate vesios povided the followig paametes apply: a) symmetical case: C = C =; µ = µ =; = + is fom to ( ad ae fom to ); fo m g i ( g = i = C () fo m g i( g = i = C () g C ( g C ) = m 3 g i ( g =.5 i = C () b) osymmetical case: C =5 C =5; othe paametes ae idetical to the pevious case. he diagam of elatioship betwee the loss pobability ad the icomig taffic desity fo the homogeeous ad heteogeeous models of the symmetical case is show i Figues -4. fo E+ E- E- E-3 E-4 E-5 E-6 E-7 E Model _ Model _ Model 3_3 Figue. Relatioship betwee ad fo models m _ l = 3 case a). l m l E+ E- E- E-3 E-4 E-5 E-6 E-7 E-8 E Model _ Model _3 Model Figue 3. Relatioship betwee ad fo heteogeeous models case a). E+ E- E- E-3 E-4 E-5 E-6 E-7 E-8 E Model _3 Model _ Figue 4. Relatioship betwee ad fo homogeeous ad heteogeeous models case a). As expected a icease i the icomig taffic desity is followed by a icease of the loss pobability. As show i Figue the geate loss pobability coespods to the _ model egadless of icomig taffic desity. It is caused by the loss i the system that becomes possible as soo as all the agets of ay goup ae busy egadless of idle devices availability i aothe goup. he smallest values of the loss pobability coespod to the _ model. Hee call ediectio is implemeted oly whe all agets ae busy i G o G. I the 3_3 model calls ca be ediected to aothe goup eve if oe device is busy. his eables a additioal load to be tasmitted to a eighboig goup.

4 8 Efimushki Žepi< hus i tems of the loss pobability the most effective amog the homogeeous models is the _ model. As fo the heteogeeous models (see Figue 3) the geatest loss pobability occus i the _3 model. I the _ ad _3 models the taffic is ediected to G whee the edistibutio mechaism is ot implemeted. his causes a icease i the loss pobability. As taffic ediectio ca be implemeted ealie ad moe itesively i m3 tha it happes i m the the loss pobability which coespods to the _3 model is highe tha that of the _ model. he least loss pobability coespods to the _3 model. It is caused by the edistibutio mechaism implemeted i both goups ad the loss becomes possible oly afte all the agets ae busy i each of the goups. Compaig the loss pobabilities fo the _ ad _3 models (see Figue 4) we ca see that i case of low ad medium icomig taffic desity ( ) the loss pobability i the _ model is highe tha that of the _3 model but i case of high icomig taffic desity ( 4 ) the loss pobability i the _ model appeas to be lowe tha that of the _3 model. I case of low load a elatively small call flow is ediected fom G to G i the _3 model but i case of a high load the umbe of ediected calls iceases ad causes a icease i the loss. he diagams of elatioship betwee the loss pobability ad the icomig taffic desity fo homogeeous ad heteogeeous models of the osymmetical case ae show i Figues 5-9. I this case the i _ ad _ i models ae diffeet ad the diagams fo compaig the loss pobability ae give fo both vaiats of models. As show i Figues 5-6 the loss pobability fo the _3 ad _ models sigificatly exceeds the loss pobability fo the 3_ ad 3_ models. his is caused by the fact that i the fist case (_3 ad _) the taffic comes fom aothe goup to a goup whee the edistibutio mechaism is ot implemeted ad the umbe of agets is smalle. E+ E- E- E-3 E-4 E-5 E-6 E-7 E Model _ Model _ E+ E- E- E-3 E-4 E-5 E-6 E Model 3_ Model _3 Fig. 6. Relatioship betwee ad fo models _3 3_ case b). I case of low ad medium icomig taffic desity the loss pobability i the _3 model is lowe tha that of the 3_ model (see Figue 7) but i case of high icomig taffic desity the loss pobability i the _3 model exceeds the loss pobability i the _3 model. E+ E- E- E-3 E-4 E-5 E-6 E-7 E-8 E-9 E Model _3 Model 3_ Figue 7. Relatioship betwee ad fo models 3 3 case b). Hee the most effective amog the homogeeous models is the 3_3 model (see Figue 8). Similaly to the symmetical case this ca be explaied though chaacteistics of the edistibutio mechaism implemeted i m 3. Compaig the loss pobability fo the 3_3 3_ 3 _3 models (see Figue 9) we ca coclude that i case of a low ad medium icomig taffic desity the least values of the loss pobability coespod to the _3 model ad i case of a high icomig taffic desity they coespod to the _3 model. It should be oted that i case of high values of the diffeece betwee the loss pobability values fo the 3_ ad 3_3 models is small (lowe tha.). Figue 5. Relatioship betwee ad fo models case b).

5 Pefomace evaluatio of Call-cete with call ediectio 83 E+ E- E- E-3 E-4 E-5 E-6 E-7 E Model _ Model _ Model 3_3 Figue 8. Relatioship betwee ad fo homogeeous models case b). E+ E- E- E-3 E-4 E-5 E-6 E-7 E-8 E-9 E Model 3_3 Model _ Model _3 Model 3_ Model 3_ Figue 9. Relatioship betwee ad fo models 3_3 3_ 3 _3 case b). Accodig to Figue 9 fo the case of low ad medium we ca coclude that if m3 is implemeted i a goup with a smalle umbe of agets the the loss pobability is geate compaed to simila models whee m is implemeted i a goup with a smalle umbe of agets. Howeve i case of a high taffic desity smalle values of coespod to models whee m3 is implemeted i G. It may be cocluded that i case of implemetig heteogeeous models i coditios of a high icomig taffic desity it is advisable to implemet m3 i bottleecks. 4 Coclusio he poposed aalytical model of Call-cete opeatio with ediectio of calls betwee goups of agets eables a ivestigatio i diffeet call taffic maagemet schemes fo the cases of agets oveload as well as fo the cases of agets udeload. he umeical aalysis poved effectiveess of the applied call ediectio pocedue. 5 Refeeces [] D. James A. Kwiatkowsk Custome Iteactio Maagemet: Developig Call Cetes ito Cotact Cetes Ovum Lodo. [] Mlaka Mate Call cetes ad goupwae systems i ifomatized sevice ogaizatios Cele. [3] A. Madelbaum Call Cetes Reseach Bibliogaphy with Abstacts V.5: July [4] V. Efimushki D. Jepic Pefomace aalysis of Cotact-Cete with goup e-allocatio of agets // It. Symposium o elecommuicatios VIEL 4 «Next Geeatio Use» May Maibo Sloveia. [5] W. Felle A itoductio to pobability theoy ad its applicatios Wiley Seies i Pobability ad Mathematical Statistics V Vladimi Efimushki eceived his diploma i mathematics fom the PFUR Uivesity of Moscow Russia i 98 ad his Ph.D. degee i theoetical basics of ifomatics fom the Istitute of Ifomatio asmissio Poblems of the Russia Academy of Scieces i 989. Now he is with the Cetal Sciece Reseach elecommuicatio Istitute (ZNIIS) of the Miisty of Ifomatio echologies ad Commuicatios of Russia as a diecto fo R&D. He is a visitig pofesso at the elecommuicatio Systems depatmet of PFUR ad vicehead of the elecommuicatio echologies ad Sevices depatmet of the Moscow echical Uivesity of elecommuicatios ad Ifomatics. His mai eseach iteests iclude achitectues potocols ad QoS i telecommuicatio etwoks pefomace aalysis modelig ad queueig systems. He is a autho of ove scietific ad techical papes. He is a academicia of the Iteatioal elecommuicatio Academy. Dago Žepi gaduated fom the Faculty of Electical Egieeig Uivesity of Lublaa i 976. He is employed with Iskatel whee he has woked i developmet of hadwae ad softwae fo the SI switchig system (as a system egiee) S telecommuicatios system ad i EWSD system (as a poect maage). He is a membe of echical Sales whee he deals with geeal telecommuicatios issues icludig QoS. He is a academicia of the Iteatioal elecommuicatio Academy.